Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is 3-connected plane graph with n vertices, then the number of colors in such a coloring does not exceed ⌊ 7n−8 9 ⌋ . If G is 4-connected, then the number of colors is at most ⌊ 5n−6 8 ⌋ , and for n ≡ 3 (mod 8), it is at most ⌊ 5n−6 8 ⌋ − 1. Finally, if G is...

Colorings Of Plane Graphs With No Rainbow Faces

Vertex rainbow colorings of graphs

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

commuting and non -commuting graphs of finit groups

فرض کنیمg یک گروه غیر آبلی متناهی باشد . گراف جابجایی g که با نماد نمایش داده می شود ،گرافی است ساده با مجموعه رئوس که در آن دو راس با یک یال به هم وصل می شوند اگر و تنها اگر . مکمل گراف جابجایی g راگراف نا جابجایی g می نامیم.و با نماد نشان می دهیم. گرافهای جابجایی و ناجابجایی یک گروه متناهی ،اولین بار توسطاردوش1 مطرح گردید ،ولی در سالهای اخیر به طور مفصل در مورد بحث و بررسی قرار گرفتند . در ،م...

Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph G is strongly rainbow-connected. The minimum number of colors needed to make G rainbow-connected is known as the rainbow connection number...